On the expected value of energy in groups

TL;DR

This paper derives bounds and asymptotic formulas for the expected energy of random subset pairs in permutation groups, with applications to growth properties and set cardinalities.

Contribution

It provides explicit bounds and asymptotic results for the expected energy in permutation groups, extending understanding of subset behavior and growth.

Findings

Derived explicit upper and lower bounds for expected energy.

Obtained sharp asymptotic formulas for specific subset pairs.

Applied results to probabilistic growth and set cardinality comparisons.

Abstract

We obtain explicit upper and lower bounds for the expected action energy associated with a pair $({\sf A},{\sf \Delta})$ of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form $({\sf A},{\sf A})$ and $({\sf A},{\sf A}^{-1})$. Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets $|{\sf A}^{\ast 2}|$ and $|{\sf A}{\sf A}^{-1}|$.